Degenerate parametric curves

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Abstract

This paper discusses two degenerate cases of polynomial parametric curves for which the degrees of the defining polynomials can be reduced without altering the curve. The first case is the improperly parametrized curve for which each point on the curve corresponds to several parameter values. The second case, which can only occur for rational polynomial parametric curves, exists when the defining polynomials all have a common factor.

This paper describes how to detect and correct each type of degeneracy. Examples are given which demonstrate that seemingly innocuous Bézier curves may suffer from either of these degeneracies.

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    According to Lemma 2, for a prime n a red curve never appears in the nth column. In consequence, no improperly parameterized quadratics (n = 2) or cubics (n = 3) exist, aside from straight lines, as Sederberg [15] already noted. In contrast, there exist improperly parameterized quartics (n = 4), from parameter substitution of degree r = 2 in a quadratic (m = 2).

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This research was performed under a grant from General Electric Company.

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